Questions tagged [numerics]

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1answer
37 views

Type of Rosenbrock method by its coefficients

A Fortran code that solves stiff PDE systems contains the following arrays of Rosenbrock-Wanner method coefficients: ...
2
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0answers
41 views

evaluating $\coth(x) - 1/x$ for real x, on 2 “pieces”

The function $\coth(x) - 1/x$ has a removable singularity at 0. Its Taylor series is: $$ \coth(x) - 1/x = \frac{x}{3} - \frac{x^3}{45} + \frac{2x^5}{945} + \ldots $$ I would like to evaluate the ...
2
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1answer
48 views

pdepe or Crank-Nicolson? How much is pdepe good?

I am beginner in MATLAB and similar. I sow and discussed with my professors doing simulations some times: they wrote down a lot of calculus, most of them using Crank-Nicolson Method and so implement ...
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0answers
52 views

Comparison of diffusion time - theoretical value vs computed

This is a follow up to my previous post I've been trying to compare the diffusion time obtained from theoretical derivation(answered in my previous post) and what is obtained computationally, for a ...
2
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1answer
100 views

mesh dependence of numerical adjoint solution

I am solving the steady, two-dimensional adjoint Euler equations, $$A_x^T \partial_x \Psi + A_y^T \partial_y \Psi = 0$$, where $A_x = \partial F_x/\partial U$ and $A_y= \partial F_y/\partial U$ are ...
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2answers
167 views

What are the most important theorems in computational science? [closed]

I was reading the book: The Finite Element Method: Theory, Implementation, and Applications by Mats Larson and Fredrik Bengzon, in page 140 of this book they say this: "The Lax-Milgram Lemma is one ...
2
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0answers
33 views

How to account for a corner node with zero-flux condition at an extrapolated distance

I am trying to implement a numerical solver and am having troubles dealing with boundary conditions, especially in the corners. I have a 2D mesh, and on the left I have a Dirichlet condition, on the ...
14
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1answer
953 views

Conserving Energy in Physics Simulation with imperfect Numerical Solver

I am creating a C++ Physics Simulation where I need to move an rigid body through an acting force field. Problem: simulation does not conserve energy. Quesiton: abstractly, how is conservation of ...
2
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0answers
27 views

Computing convolution of two characteristic function over a 1D Cartesian mesh

I am trying to compute the convolution of two characteristic functions over a Cartesian mesh. First, I define my Cartesian mesh of the interval $[0,1]$ as follows $$ x_{i} = i \Delta x, i = 0, 1, 2\...
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0answers
137 views

Ising model simulation offset critical temperature and interal ernergy

I'm writing a code for the Ising model using WHAM (the weighted histogram analysis method),But it seems to produce critical temperature and internal energy wrong. (newest rewritten code is below) <...
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0answers
53 views

Stably solve transport equation with source term

I am trying to solve a transport equation of the form for the variable $\psi(t,r)$ \begin{equation} \partial_t\psi-\alpha(r)\partial_r\psi-\beta(r)^2\psi-f(t,r)=0 , \end{equation} where I am solving ...
0
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1answer
82 views

Solution of thermal analysis using finite element

I want to solve a thermal analysis using finite elements. The governing equation is $$C \frac{dT}{dt}+K T = Q$$. When using backward differencing for time, the resulting equation is quite straight ...
4
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1answer
81 views

Modelling flow through pipe networks

I'm trying to educate myself on modelling solute flows through pipe networks. This is a follow up of my previous post here $$\frac{\partial C}{\partial t} = - v\frac{\partial C}{\partial x}$$ While ...
2
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1answer
82 views

Does mass balance hold in convective diffusion

I'm trying to understand how convection-diffusion equations are solved in pipe flow modules available in CFD solvers. $$ \frac{\partial C}{\partial t} + \nabla \cdot (\mathbf{v} C) = \nabla \cdot (D \...
0
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1answer
67 views

Solving differential equation in Python with discretized variable coefficients

I am trying to solve a differential equation with discretized variable coefficients which are calculated from a time serie. In this case the Runge-Kutta step size is fixed by the frequency in the time ...
3
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0answers
65 views

Numerical integration with singularity term

In https://www.johndcook.com/blog/2012/02/21/care-and-treatment-of-singularities, the author explains the subtraction method to get rid of singularities when performing numerical integration. The ...
6
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2answers
1k views

Runge-Kutta in the presence of an attractor

Suppose you are solving a system of equations numerically that possesses an attractor (no matter the initial conditions set, all the different solutions will approach a specific set of values that ...
6
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0answers
97 views

How to check if my stiffness matrix is correct

I built the stiffness matrix for the Poisson equation on a 2-dimensional domain with the shape of "almost" an octagon, using pyramid basis functions. I used almost to intend the fact that I have an "...
5
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2answers
123 views

Is the diffusion equation with Neumann and Dirichlet BCs well-posed?

I am considering the following diffusion equation: $$\frac{\partial f}{\partial t} = \frac{\partial}{\partial x}[D(x,t)\frac{\partial f}{\partial x}]$$ over a grid ...
2
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1answer
43 views

Numerical integration of the dataset of a function

The energy equation for a spherically symmetric system is given by $$\mathscr{E}=\frac{v^2(r)}{2}+\frac{c_s^2(r)}{\gamma-1}+\phi(r)$$ where $\mathscr{E}$ is the total energy, $v$ is the velocity of ...
2
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1answer
67 views

Defining Current Density in a FEM model (MATLAB)

I'm attempting to solve the Poisson equation in 3D for a magnetic vector potential in the presence of a current source. To validate my code, I'm initially looking to reproduce the model described in ...
-1
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1answer
39 views

Integrate using Composite Simpson's rule

In a question, we have been given the speed of a car at time t= 0,2,4,6,.......,20 minutes.But it asks us to approximate the distance travelled by the car in 30 minutes using Composite Simpson's rule. ...
1
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1answer
90 views

Solve linear system with Newton-Raphson method

Is it possible to solve a linear matrix system $A x = b$ using the Newton-Raphson method? If yes, how can this be done? More special, how is the derivative build?
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1answer
54 views

Recursive Algorithm to Calculate Determinant via Expansion of Minors in C#

I have been recently trying to attempt to write an algorithm in C# that would calculate the determinant of a matrix via recursion using the expansion of minors method. I understand that there are ...
2
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0answers
94 views

What exactly is the cause(s) of blow-up for too-large step size in a method like RK4?

I have been working on creating a few home-made numerical methods, and I am using them to visualize text-book problems from my Strogatz dynamics textbook. It feels like a good way to learn numerical ...
0
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0answers
22 views

Interpreting results of using no-flux boundary condition

I am solving for solute transport in 1 D. $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - v\frac{\partial C}{\partial x}$$ No-flux boundary condition is imposed at both the ...
1
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1answer
69 views

Implementing Robin Boundary condition (finite difference)

I'm interested in applying Robin boundary condition to a convection-diffusion problem in 1D. In the following system, $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2} - v\frac{\...
2
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0answers
85 views

Extracting FEM matrices in matlab pde toolbox

I am trying to follow the dynamic linear elasticity in Matlab, link here. My goal is to extract the FE Matrices using the function assembleFEMatrices in matlab and solve the resulting system of second-...
-1
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1answer
109 views

Imposing periodic boundary condition for linear advection equation - Node problem

I've spent the whole day trying to figure out what is the correct way to impose (and implement) periodic boundary conditions $u(0,t)=u(1,t)$ for all $t>0$ for the simple advection equation $u_t + v ...
0
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1answer
83 views

Robin Boundary Condition with Implicit Upwind - Finite Difference Method for 2D Convection-Diffusion Equation

I am trying to solve a problem with 2D Convection-Diffusion equation with U = Concentration ($mg/m^{2}$) using Implicit Upwind Finite Difference Method like this $$ \frac{\partial U}{\partial t} + ...
1
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1answer
118 views

Runge-Kutta fourth order method. Integrating backwards

I am using a Runge-Kutta fourth order method to solve numerically the usual equation of motion of a background scalar field in curved spacetime with a quartic potential: $\phi^{''}=-3\left(1+\frac{H^{...
1
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1answer
73 views

How to Break Coupled ODEs down to first order for Runge-Kutta

My question might seem a bit simple. I am trying to solve a system of ODEs using Runge-Kutta method. I am having difficulty breaking down the equations into a system of first order ones required ...
2
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1answer
62 views

How to robustly and numerically expand a $k$-order polynomial in two variables defined on a polygon domain?

Given a $k$-order polynomial in two variable $p(x, y)$ defined on a polygon domain $K$. And I want to numerically expand it to the following form $$ p(x, y) = c_0 + c_1 x + c_2 y + c_3 x^2 + c_4 xy + ...
1
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0answers
51 views

Numerical Method for Equation System of two depending Equation Systems

I am searching a solution method for the following equation system of equation systems: Let $A \in \mathbb{R}^{n \times n}$ be an invertible Matrix, $f, b_1, b_2 \in\mathbb{R}^n$ given vectors and $ ...
6
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1answer
117 views

Computing square root of diag(u)-uu'?

I need an efficient way to take square root of a matrix which is a sum of diagonal matrix and rank-1 matrix. More specifically it's the following matrix $$A=D-uu'=\text{diag}(u)-uu'$$ Where entries ...
1
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2answers
48 views

Numerical integral with symbolic integral in exponent

Many times in fourier approximation we come across integrals such as $$\int_0^1 e^{-\gamma\int_0^xu_0(\eta)d\eta}dx$$ where $\gamma$ is a constant and the data for $u_0$ is provided as a discretely ...
3
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1answer
53 views

Bounding error of float32 matrix multiplication

Some numerical debugging led me to the minimal example below. I'm observing relative error of 0.75 on individual elements. Is there a way to estimate/bound this error without resorting to higher ...
2
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1answer
47 views

Passing data as arguments in ODE45

I need to import data from file in order to describe the structure of a network. I used the following: weights = readtable('weights192.txt'); W = weights{:,:}; ...
2
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0answers
56 views

Numerical integration of SDE: choice of $dt$ and algorithm

I am working on the following Stochastic Differential Equation (SDE) in the Quantum Mechanics context: $$dX_{t} = a X_{t} dt + b X_{t} dW$$ where $X_{t}$ is my stochastic varible, $dt$ is my ...
1
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1answer
43 views

Weighted moving variance

i have a time-series and, in analogy with exponentially weighted moving average, i would like to compute the exponentially weighted moving standard deviation or variance in an efficient, numerically ...
2
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0answers
39 views

Inverses of many standard subspaces of one large matrix

i have a large rectangular invertible matrix M (about 5000x5000) and i have a loop in which i do the following for each iteration i (there are about 6000 iterations): i am given a subspace S_i (which ...
1
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0answers
72 views

Runge-Kutta for PID and system in separate calculations without filter

I need to calculate a closed-loop system in Python; specifically, obtain the PID response and then use the output to obtain the system response sample-by-sample with my own loop. For this, I am ...
0
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0answers
64 views

How to implement the following Finite Element method for Burgers' equation?

I am trying to replicate this result. It involves using the Galerkin finite element approach onto the viscous Burgers' equation. However, my implementation (in R) seems to be giving me wrong results....
0
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1answer
87 views

Well-posedness of Navier-Stokes equation

Before running a simulation for turbulence (e.g Rayleigh-Benard Convection), how do we check for well-posedness of Navier-Stokes with other equations for a given boundary condition?? Can someone ...
7
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1answer
122 views

Numerically estimating expected value of f(x) when x is normally distributed

I need to estimate $$ \mathbb{E}_x[f_i(x)] = \int_{\mathbb{R}^n} f_i(x) p(x) dx $$ for many functions $f_i(x)$, where $p(x)$ is the density of a normal distribution. The evaluation of all the ...
5
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0answers
48 views

Evaluate Nth root of a rational to a correctly rounded float

Excuse my lack of vocabulary for I have no formal training in this field, which is also why I ask this question - it may be trivial or it may be impossible. I want to evaluate an expression in the ...
1
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0answers
88 views

How to compute the following Crank-Nicolson scheme for the viscous Burgers' Equation?

I am attempting to replicate results from this article. I'm not sure why but my results are completely different and wrong. For example, the exact solution with parameters ($x=0.1$, $T=0.01$, $Re=0....
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0answers
79 views

What's the more efficient way to solve this matrix equation?

This is intended to be a more generic question not about a specific system. Given a hermitian matrix $H(x_1,\dots,x_n)$ depending non-linearly on some real parameters $x_1,\dots,x_n$. We want these to ...
4
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0answers
100 views

Integrators for Nonlinear/Stiff PDE

It was suggested I ask this question in this section. Anyway: I have a particular nonlinear PDE of the form $$ u_t(x,t)=iu_{xx}(x,t)+f(x,u(x,t)) \tag{1} $$ Where f is some nonlinear function. With ...
1
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0answers
48 views

WENO scheme on curvilinear coordinates

I've been developing a curvilinear FVM code. So far I've implemented the PPM scheme and am looking into adding WENO schemes. So far I've been discretizing the grid metrics using a second-order central....